A variety of analyses on MABs involving risk measures have been carried out. [1]} considered the mean-variance as their risk measure. Each arm \(i\)  followed a Gaussian distribution with mean \(\mu _i \in [0, 1]\)  and variance \(\sigma _i^2 \in [0, 1]\) . The authors provided an LCB-based algorithm with accompanying regret analyses. [2]} proposed the Multi-Armed Risk-Aware Bandit (MaRaB) algorithm with the goal of minimising the number of pulls of risky arms, using the risk measure CVaR. [3]} demonstrated that the instance-dependent and instance-independent regrets in terms of the mean-variance of the reward process over a horizon \(n\)  are lower bounded by \(\Omega (\log n)\)  and \(\Omega (n^{2/3})\)  respectively. [4]} analysed contextual bandits with risk constraints, and developed a meta algorithm which makes use of the online mirror descent algorithm that achieves near-optimal regret with respect to minimising the total cost. [5]} designed the first Thompson sampling algorithm for risk measures, particularly the mean-variance risk measure for Gaussian bandits, and proved near-optimal regret bounds under specific regimes. [6]} designed a Thompson sampling algorithm factoring a user's “risk tolerance” level, either minimising mean rewards under some “maximum risk” criterion, or simply minimising the risk measure. [7]} designed and analysed Thompson sampling-based algorithms \(\alpha \) -NPTS for bounded rewards and \(\alpha \) -Multinomial-TS for discrete multinomial distributions.
